Laser device with a convergent reflector topology

ABSTRACT

A laser device, having an optical cavity containing a gain medium, a total reflectance reflector positioned within the optical cavity, and a partial reflectance reflector positioned within the optical cavity in a juxtaposed relationship to the total reflectance reflector. The topology of the reflectors are defined by a convergent reflector topology function that converges light emitted within the optical cavity to a laser beam that exits the optical cavity through the partial reflectance reflector by a series of reflections between the total reflectance reflector and the partial reflectance reflector. The laser beam emitted from the optical cavity has a predefined pattern for a given convergent reflector topology.

CROSS REFERENCE

This application is a continuation of U.S. patent application Ser. No.13/206,201, filed Aug. 9, 2011, now U.S. Pat. No. 8,630,825, issued Jan.14, 2014, which, in turn, claims the benefit of U.S. Provisional PatentApplication No. 61/425,433, filed Dec. 21, 2010, which is incorporatedherein by reference in its entirety.

BACKGROUND OF THE INVENTION

Light Amplification by Stimulated Emission of Radiation (laser) providesfor a spatially coherent, spatially narrow, spectrally narrow(monochromatic), low divergence beam of electromagnetic radiation.Lasers can be found in a number of modern-day applications, such ascutting tools, surgery, optical storage devices, and opticalcommunications. A laser beam is produced by reflecting light back andforth using reflectors through an optical cavity containing a gainmedium having an energy population inversion (i.e. more electrons in arelatively high energy state versus at a low energy state). Thereflectors used for the reflection are typically flat, spherical, orparabolic. Additionally, once the laser beam is emitted through apartially transmitting reflector, the beam can be manipulated by opticalelements such as reflectors, lenses, polarizers, optical fiber, anddiffraction gratings. For some applications, such optical elements mayneed to be very precisely manufactured and may have a resulting highcost. Lenses, in particular may need to have a very precise shape andsurface finish for some applications. The material properties of a lens,such as the index of refraction, vary with the wavelength of the laserbeam. The uses of lenses to converge a beam of light may lead to variousoptical aberrations, including spherical and chromatic aberrations.

SUMMARY OF THE INVENTION

The invention relates to a laser device, having an optical cavitycontaining a gain medium, a total reflectance reflector positionedwithin the optical cavity, and a partial reflectance reflectorpositioned within the optical cavity in a juxtaposed relationship to thetotal reflectance reflector. The device comprises comprising: whereinthe total reflectance reflector has a first surface topology; whereinthe partial reflectance reflector has a second surface topology; whereinat least one of the first surface topology and the second surfacetopology is defined by a convergent reflector topology function thatconverges light emitted within the optical cavity to a laser beam thatexits the optical cavity through the partial reflectance reflector by aseries of reflections between the total reflectance reflector as definedby the first surface topology and the partial reflectance reflector asdefined by the second surface topology. The laser beam emitted from theoptical cavity has a predefined pattern for a given first surfacetopology and a second surface topology.

BRIEF DESCRIPTION OF THE DRAWINGS

In the drawings:

FIG. 1 is a schematic diagram of a laser device on which an embodimentof the invention can be applied.

FIG. 2A is a ray tracing diagram with a flat beam pattern applied to areflector design known in the prior art.

FIG. 2B is a ray tracing diagram with a cross beam pattern applied to areflector design known in the prior art.

FIG. 2C is a ray tracing diagram with a conjugate beam pattern appliedto the reflector design known in the prior art.

FIG. 3A is a ray tracing diagram with a flat beam pattern applied to areflector design according to the invention.

FIG. 3B is a ray tracing diagram with a cross beam pattern applied tothe reflector design of FIG. 3A.

FIG. 3C is a ray tracing diagram with a conjugate beam pattern appliedto the reflector design of FIGS. 3A and 3B.

FIG. 4 is a flow diagram illustrating a method of determining aconvergent reflector topology.

FIG. 5 is an example of a forcing function that provides a convergentreflector design.

FIG. 6 shows a first iteration of a reflector topology based upon themethod of FIG. 4.

FIG. 7 shows a subsequent iteration of a reflector topology based uponthe method of FIG. 4.

FIG. 8 is a flow diagram for determining a forcing function that willgenerate a convergent reflector topology when applied to the method ofFIG. 4.

FIG. 9 is a fractal diagram of a forcing function that does not havenon-convergent areas and will not generate a convergent reflector designwhen applied to the method of FIG. 4.

FIG. 10 is a fractal diagram of a forcing function determined accordingto the method of FIG. 8 such that when the forcing function is appliedto the method of FIG. 4, a convergent reflector topology results.

FIG. 11A is a schematic representation of a laser beam pattern that canbe generated using known reflectors in the prior art.

FIG. 11B is a schematic representation of another laser beam patternthat can be generated using reflectors designed using the methoddescribed in FIG. 4.

FIG. 11C is a schematic representation of yet another laser beam patternthat can be generated using reflectors designed using the methoddescribed in FIG. 4.

FIG. 11D is a schematic representation of yet another laser beam patternthat can be generated using reflectors designed using the methoddescribed in FIG. 4.

FIG. 12A is a schematic representation of a laser beam pattern that canbe generated by using the method described in FIG. 4 and then bemodified to generated a single emission point.

FIG. 12B is a schematic representation of a laser beam pattern with asingle emission point by modifying the reflector design of FIG. 12A.

FIG. 13 is a flow diagram illustrating a method of modifying aconvergent reflector design to achieve a modified emission pattern.

DESCRIPTION OF AN EMBODIMENT OF THE INVENTION

The present invention relates generally to a method of determining thetopology of reflectors that provide convergence of light that reflectsupon it. More specifically, the topology of a convergent reflector isdetermined by using an iterative algorithm for converging upon thedesired topology. The embodiments described herein can be applied to anyconvergent reflector design.

Although the convergent reflector designs are described in the contextof optics and more particularly laser cavities, the convergent reflectordesigns can apply to any type of reflector used for reflecting any typeof radiation or matter.

FIG. 1 shows a schematic diagram of a laser device 10 for producing aspatially coherent, spatially narrow, spectrally narrow (monochromatic),low divergence laser beam 18 of electromagnetic radiation as known inthe prior art. A laser device 10 comprises an optical cavity 16containing a gain medium 24 with a total reflectance reflector 12 on oneend and a partial reflectance reflector 14 on the other end. The laserbeam 18 is emitted through the partial reflectance reflector 14. A lens20 may be used to focus the laser beam 18 into a focused beam 22.

The laser device 10 can be any one of various types, including, but notlimited to solid state, gas, excimer, and semiconductor lasers. Thesevarious types of lasers differ in the type of gain medium 24 containedwithin the optical cavity, as well as, the methods of delivering powerto the gain medium. For example, the gain medium in a gas laser containsa gas such as a mix of Helium and Neon and a solid state laser may havea gain medium comprising Neodymium-doped yttrium aluminum garnet(Nd:YAl₅O₁₂ or Nd:YAG). To produce laser radiation, the gain medium 24needs to be pumped, or supplied with energy to achieve a state called apopulation inversion, where there are more electrons at a higher energylevel than a lower energy level. As electrons lose energy from thehigher energy level to the lower energy level, the lost energy isemitted as a photon. If some of the light produced in the optical cavityis fed back into the cavity, by reflecting the light off of both thetotal reflectance reflector 12 and the partial reflectance reflector 14,stimulated emission and coherence of the light is achieved. Thereforethe reflectors 12 and 14 are critical to the efficient functioning ofthe laser device 10.

The total reflectance reflector 12 reflects substantially all the lightthat is incident upon it. The partial reflectance reflector 14 reflectsmost of the light that is incident upon it and allows some of the lightto transmit through the reflector 14. For example, a partial reflectancereflector 14 may allow 3% of light to pass and 97% reflectance.Reflectors 12 and 14 enclosing the optical cavity 16 in the current artare typically flat, parabolic, or spherical shaped. Such conventionalreflectors may result in reflections that do not converge the light.Reflectors 12 and 14 that converge the light rather than diverge thelight within the optical cavity 16 may provide the advantage of greaterlasing efficiency. Additionally, it is important to note that the use ofconvergent reflectors may pose the advantage of not requiring a lens 20outside of the laser device 10 to shape the laser beam 18 emitted fromthe laser device 10. Not requiring an external lens 20 or other opticalelements can present a cost, form factor, and performance advantage inmany systems and applications.

FIGS. 2A, 2B, and 2C are ray tracing diagrams of flat, cross, andconjugate beam patterns, respectively with conventional reflector designfor both the total reflectance reflector 50 and the partial reflectancereflector 52. FIG. 2A shows that with a flat beam 56, which may reflectback and forth multiple times between the two reflectors 50 and 52, thestart and end points are at the same distance away from the optical axis54 and is emitted through the partial reflectance reflector 52 as beam58. With the cross beam pattern of FIG. 2B, the same phenomenon arises,where the cross beam 60 starts and ends at the same distance away fromthe optical axis 54 after one or more reflections and is emitted throughthe partial reflectance reflector 52 as beam 62. Again the beams 72emitted through the partial reflectance reflector 52 are at the samedistance away from the optical axis 54 as where the beam 70 started inthe case of conjugate beam reflections of FIG. 2C. Therefore, forconventional reflector designs, there is no convergence in the beam,since the start locations and the end locations of the beams are thesame distance away from the optical axis 54 with symmetry around theoptical axis 54. In other words, with conventional reflector designs,the optical beam, if started at a particular distance away from theoptical axis 54, do not converge upon a point that is a differentdistance away from the optical axis 54 after multiple reflections bydesign.

With conventional reflectors, such as flat, spherical, and parabolicshapes, the emitted beam from the partially transmitting reflector isideally the same distance away from the optical axis as where theoptical beam started. However, there may be non-idealities such assurface imperfections on the reflector or alignment errors of thereflectors which may induce divergence of the beam after multiplereflections, where the optical beam is emitted at a distance from theoptical axis that is greater than the distance where the optical beamstarted. As a result, in practical implementations of known reflectortopologies 50 and 52, the electromagnetic radiation emitted aftermultiple reflections between the two reflectors may result in divergenceof the optical beam.

In contrast to the non-converging reflectors of 2A-C, a convergentreflector behavior is shown in FIGS. 3A-C, where a light beam's enddistance relative to the optical axis 130 is less than the startposition relative to the optical axis 130. Therefore, the lightconverges to one or more points.

FIGS. 3A, 3B, and 3C are ray tracing diagrams of flat, cross, andconjugate beam patterns, respectively of a convergent reflector designfor both the total reflectance reflector 120 and the partial reflectancereflector 122. FIG. 3A shows that with a flat beam 136, which mayreflect back and forth multiple times, such as at 128 between the tworeflectors 120 and 122, as multiple reflected beams, such as beam 132,the start point may be at the start beam 136 and the end beam may be atthe optical axis 130 and emitted through the partially reflectingreflector 122 as beam 138 and 140. As a result the beam converges to apoint relative to the optical axis 130 on the output of the opticalcavity that is different from the start point relative to the opticalaxis 130. With the cross beam pattern of FIG. 3B, the same phenomenon asFIG. 3A arises, where the cross beam starts and ends at different pointsrelative to the optical axis 130. In the case of the cross beam forexample a beam that starts at point 154 may converge at two points 154and at the optical axis 130 and emitted as beams 156 and 158 through thepartially transmitting reflector 122 after multiple reflections. Againthe beams 164 and 166 emitted from the partial reflectance reflector 122are at a different distance away from the optical axis 130 compared towhere the beam 160 started in the case of conjugate beam reflections ofFIG. 3C. Therefore, for convergent reflector designs, there isconvergence in the multiple reflected beams, since the start locationsand the end locations of the beams are different distances away from theoptical axis 130. In other words, with convergent reflector design, theoptical beam if started at a particular distance away from the opticalaxis 130 may converge upon a point that is a different distance awayfrom the optical axis 130 after multiple reflections by design.

It is seen by comparing the ray tracing diagrams of FIGS. 2A, 2B, and 2Cto FIGS. 3A, 3B, and 3C, the beams can be converged within the opticalcavity by a series of reflections and can be emitted from the opticalcavity in a predefined pattern without the use of optical elementsoutside of the optical cavity. A method is disclosed herein that can beused to derive the topology of the reflectors. In particular the methodcan be used to derive reflector topologies that result in convergence ofa laser beam. Convergence within the optical cavity, instead ofrequiring optical elements outside of the optical cavity can poseseveral advantages including reduced cost, reduced form factor andgreater efficiency and reliability.

FIG. 4 is a flow diagram illustrating the method of determining thesurface topology of a reflector surface 200. First a forcing function,f(x), is determined at 202. The choice of forcing function is veryimportant in the determination of the surface topology. To have aconvergent design, the forcing function, f(x), must be one that hasareas that do not converge in the x+jy space. This will be described ingreater detail in conjunction with FIGS. 8, 9, and 10. Once a forcingfunction is determined at 202, the forcing function is applied to aconvergent algorithm to generate y values of the reflector topology at204. The convergent algorithm can be any known algorithm for iterativedetermination of roots of a function. For example one such convergentalgorithm can be the Newton-Raphson algorithm described as:

$x_{n + 1} = {x_{n} - \frac{f(x)}{f^{\prime}(x)}}$

where, x_(n+1) is the approximation of the value of a root of a functionderived upon iterating the convergence algorithm one time beyond theapproximation x_(n), f′(x) is the first derivative of the forcingfunction f(x), and n is a real integer. Although the Newton-Raphsonalgorithm is used throughout this specification, any known convergencealgorithm may be used.

The set of x values, x_(p), generated by subjecting the forcing functionf(x) to the convergent algorithm is the set of y-values in the x-y spacein which the reflector topology is defined. In other words, thenumerical values of {x₁, x₂, . . . , x_(p)} in the x+jy space of theforcing function are numerically equivalent to y₁, y₂, . . . , y_(p) inthe x-y space in which the surface topology of two reflector surfacesare defined as a set of points with p members. The reflector surface isdefined as set of points in the x, y space, with p members. For purposesof discussion, the complete set of x values, {x₁, x₂, . . . , x_(p)} inthe x-y reflector topology space is defined as x_(P). Similarly, thecomplete set of y values, {y₁, y₂, . . . , y_(P)} in the x-y reflectortopology space is defined as y_(P). Therefore the reflector surface canbe defined as a set of paired points {x_(P), y_(P)} with p members.

At 204, the x-values, x_(P), of the surface topology are known as aresult of the size of the optical cavity. In other words, if the opticalcavity has a length of λ, then the x-values may be either −λ/2 or λ/2 onthe initial iteration. At 206, the slope at each y_(P) is determined togenerate a set of y values and slopes {y_(P), m_(P)}, where m_(P) is aset of slope values, {m₁, m₂, . . . , m_(p)} corresponding to each ofthe x-values, x_(P) and y-values, y_(P). The slope is calculated at anypoint, i, in the set of y_(P), by considering the values of x_(i),y_(i), x_(i−1), y_(i−1), x_(i+1), and y_(i+1). In other words, the slopeat any point i, m_(i), is a function of the coordinates of that point(x_(i), y_(i)), as well as the coordinates of the previous (x_(i−1),y_(i−1)) and subsequent (x_(i+1), y_(i+1)) points in the set {x_(P),y_(P)}. m_(i) is a real scalar that must satisfy a condition where abeam impingent on point (x_(i), y_(i)) with a trajectory defined by thedirection from (x_(i−1), y_(i−1)) to (x_(i), y_(i)) must be reflected tothe point (x_(i+1), y_(i+1)). The slope at any point (x_(i), y_(i)) canbe determined as:

$m_{i} = {\tan\lbrack {{\frac{1}{2}( {{\tan^{- 1}( \frac{y_{i + 1} - y_{i}}{x_{i + 1} - x_{i}} )} - {\tan^{- 1}( \frac{y_{i} - y_{i - 1}}{x_{i} - x_{i - 1}} )}} )} + \frac{\pi}{2}} \rbrack}$

It should be noted that the derivation of the slope is just one knownderivation. Any known algorithm or method may be used to determine theslope, m_(i), based on the reflections desired at that point (x_(i),y_(i)).

Next, at 208, R′(x) or the first derivative of the surface topology isdetermined by fitting to the set of y-values and slopes, {y_(P), m_(P)}.The fitting may be by any known method such as linear regression andleast squares fit. Next R′(x) is integrated to generate the surfacetopology R(x) at 210 as:R(x)=∫R′(x)*dx+K

where K is a constant of integration. If the system is centered at x=0,meaning the two reflectors are equidistant from the y-axis, then K=0.

Using the new R(x) function, new values of x are generated, x_(P),corresponding to the y-values, y_(P), at 212. Next it is determined ifthe current iteration of the set x_(P) is different from the previousset of x_(P) by a predetermined value at 214. If it is, then R(x) andtherefore {x_(P), y_(P)} of the reflector topology have not convergedand therefore the method loops back to 206, where the new x-values,x_(P), determined at 212 are used to determine new values of slope,m_(P), at each y-value at 206. Once the y-values are determined based onthe forcing function f(x) at 204, the set of y-values do not change.Therefore, at each iteration, only the x-values change based on arevised set of slopes, m_(P), with the fixed set of y-values, y_(P).After a sufficient number of iterations with the feedback loop to 206,the set of x-values, x_(P), eventually converge and the change in x_(P)from a previous x_(P) drops to less than a predetermined threshold at214. At this point, R(x), and the resulting {x_(P), R(x)} or {x_(P),y_(P)} defines the reflector surface topology.

The sequence of steps depicted is for illustrative purposes only, and isnot meant to limit the method 200 in any way as it is understood thatthe steps may proceed in a different logical or sequential order anddifferent, additional, overlapping, or intervening steps may be includedwithout detracting from the invention.

In this algorithm 200, the more data points that are used, p, the moreprecise the reflector surface topology determination will be.Additionally, more iterations at 206 can result in greater precision andaccuracy of the algorithm. The number of iterations can be increased bysetting a lower predetermined threshold for the change in x_(P).

Due to the volume of calculations required in this method 200, it isenvisioned that the method 200 is executed on a computer with electronicmemory running a mathematical software or with a software dedicated tothe execution of this algorithm. Examples of commercial software onwhich the method 200 may be implemented include Microsoft Excel®,Maplesoft Maple®, or Mathworks Matlab®.

The use of the method of determining a surface topology of a reflectorsurface 200 will be better understood by example in conjunction withFIGS. 5, 6, and 7. The top graph of FIG. 5 shows a forcing function,represented as f(x+jy) in a complex space (x+jy, f(x+jy)). To determinea convergent reflector design, f(x+jy) must have areas in the x+jy spacewhere f(x+jy) applied to a convergent algorithm does not converge to aroot of f(x+jy). In this example, P1, representing (x₁₊jy₁, f₁(x₁+jy₁))is a point within a non-convergent area and as a result in thenon-convergent area in the x+jy space is applied to the convergencealgorithm. This generates a series of points, P₂, P₃, P₄, P₅, P₆, P₇,and P_(g) in the x+jy complex space. Each of the x values of each of thepoints are set as the y values in the real x-R(x) space of the reflectortopology function. For simplicity, the method is depicted with only 8points, or 4 points on each reflector. In the first iteration the leftreflector topology is depicted as 230 and the right reflector topologyis represented as 232. As depicted in FIG. 6, in the first iteration,the reflector topology is flat, because the x values are one of twovalues for each reflector 230 and 232 on either side. Next the slopes ateach of the (x, y) points is determined at 206 so that a reflection tothe next (x, y) point is achieved along the reflection paths as depictedin FIG. 6. For example, at a certain slope at (x₂, y₂), the beamincident upon that point from point (x₁, y₁) will reflect from thesurface at point (x₂, y₂) and reach point (x₃, y₃). In effect, slope atthat point, m2 is determined such that the beam reflects from (x₁, y₁)to (x₂, y₂) and then to (x₃, y₃).

Once the slope at each of the points, p, is determined, the complete setof x-values and slopes, {x_(P), m_(P)} are fit to one of two differentfitting functions R₁′(x) and R₂′(x) for the reflectors 230 and 232,respectively. Each of the fitting functions R₁′(x) and R₂′(x) are thenintegrated to derive R₁(x) and R₂(x), respectively. New values of x,x_(P) are generated from R₁(x) and R₂(x) at each value of y_(P). FIG. 7shows the graphical representation of {x_(P), y_(P)} with the new set ofx_(P) values, where the x_(P) values are different from the x_(P) valuesin FIG. 6, and the y_(P) values are the same.

It is then determined if the x_(P) have converged by comparing the newx_(P) values to the old x_(P) values. There are many ways to determineif the functions have converged, including determining an error value ofthe average square root of the square of the difference by:

${\delta = {\frac{1}{p}{\sum\limits_{j}^{p}\;\sqrt{( {x_{j}^{new} - x_{j}^{prev}} )^{2}}}}},$

where 6 is error value, p is the total number of points, j is an index,x_(j) ^(new) is the new x value at the jth index, and x_(j) ^(prev) isthe previous x value at the jth index from the previous iteration.

This error value may be compared to a predetermined threshold todetermine if the set of x_(P) have converged. As an alternative, theerror value, δ, may be normalized by the mean of the set of x_(P), todetermine a percentage error for comparison to a predetermined thresholdof percentage error. For example the percentage error threshold may be0.1%, meaning if the difference between the new x_(P) and previous x_(P)is less than 0.1%, then the functions R₁(x) and R₂(x) are deemed to beconverged.

If it is determined that x_(P) has converged then the topology of FIG. 7represents the final topology of the reflectors 240 and 242. If however,it is determined at 214 that the values of x_(P) have not converged,then the method repeats the previous steps starting from determining theslopes at each y-value, y_(P), using the new x_(P) values at 206.

The reflector topologies determined by the method 200 can be used tofabricate reflectors 12 and 14 of the laser device 10 by any knownmethod including, but not limited to machining or three-dimensional (3D)printing. The reflectors with the determined topologies therefore are animprovement over currently used reflectors as the new reflectors canresult in greater lasing efficiency and can produce laser emissions withbeam patterns that can be formed with a reduced number of, and in somecases no additional, optical elements outside of the laser device 10.

From the discussion above, it is clear that the choice of the forcingfunction is important in determining a topology of the reflector thatleads to convergence of light reflecting therefrom. FIG. 8 is a flowdiagram showing a method of determining an appropriate forcing function230 that will produce a convergent reflector design when applied to themethod for determining a reflector topology 200 of FIG. 4. In essencethe required forcing function is one that produces areas in the x+jyspace where the function does not converge when applied to a convergentalgorithm. The method starts by picking a forcing function at 232. Theforcing function must be a third or higher order polynomial function andhave at least one inflection point. An inflection point for this purposeis a point where the second derivative of the function is zero and hasdifferent signs on either side of the inflection point. The forcingfunction is more ideally a third order or higher polynomial. Such thirdand higher order polynomials are likely to produce forcing functionsthat are non-convergent over an area in the x+jy space. The next step isto repeatedly apply a convergent algorithm to f(x) at 234. Theconvergent algorithm can be any known convergent algorithm, includingthe Newton-Raphson convergent algorithm as discussed above. Theconvergent algorithm is applied repeatedly by using a computer such as amicro-computer and the data points in the x+jy space are stored in anelectronic memory of the computer. The convergent algorithm is appliedto various portions of the x+jy space to test the convergence in alllocation. That means that for a function there are as many convergentareas as there are roots for a function. For example a fourth orderpolynomial function may have four roots, so in that case there may befour sets of areas that converge to four different roots. The areasassociated with each of the roots can be represented as such on a plotof the x+jy space. The (x+jy, f(x+jy)) data is plotted as aNewton-Raphson fractal at 236. It is then determined if theNewton-Raphson fractal has any non-convergent areas at 238. If there areno non-convergent areas in the Newton-Raphson fractal, then the functionis perturbed and the method loops back to 234 to apply the new functionrepeatedly to the convergent algorithm. If it is found that the functionf(x) does have at least one non-convergent area at 238 then the forcingfunction, f(x) can be used to generate a convergent reflector topologyand therefore an appropriate forcing function is identified at 242.

An example of a non-convergent forcing function is:ƒ(x)=x ⁴+112

Applying the above function to the method 200 of FIG. 4 will generate anon-convergent reflector topology.

An example of a convergent surface topology function is:ƒ(x)=−x ⁴+4.29x ²+5.62

Applying the above function to the method 200 of FIG. 4 will generate aconvergent reflector topology. Although in this example, f(x) is afourth order polynomial, f(x) can be a polynomial function of any order,as well as, a sinusoidal, exponential, logarithmic, or any variety ofknown functions.

The perturbation of f(x) at 240 may in most cases be by changing thef(x)-offset of f(x). For example, consider a general function of theform:

${{{f(x)} = {\sum\limits_{q}^{Q}\;{K_{q}*x^{q}}}}}_{Q \geq 3}$

where q and Q are integers and K_(q) are real numbers.

If the Newton-Raphson fractal generated from this function f(x) does notindicate non-convergent areas at 238, then the function may be perturbedby incrementing the value of K₀ by a predetermined amount. This has theeffect of increasing the f(x)-offset of the function and for manypolynomial functions, such an offset may lead to a new function with anon-convergent area in the x+jy space as indicated by areas ofnon-convergence in the associated Newton-Raphson fractal. Such functionscan then be used in the method 200 of FIG. 4 to determine a convergentreflector topology.

As a more specific example, consider a forcing function:f(x)=ax ^(b) +cx ^(d) +e,

where b and d are integers and a, c, and e are real numbers.

If the Newton-Raphson fractal generated from this function f(x) does notindicate non-convergent areas at 238, then the function may be perturbedby incrementing the value of e by a predetermined amount. This has theeffect of increasing the f(x)-offset of the function and for manypolynomial functions, such an offset may lead to a new function with anon-convergent area in the x+jy space as indicated by areas ofnon-convergence in the associated Newton-Raphson fractal. Such functionscan then be used in the method 200 of FIG. 4 to determine a convergentreflector topology.

FIG. 9 shows an example of a Newton-Raphson fractal without areas ofnon-convergence. As a result the function of this fractal is not anappropriate forcing function to generate a convergent reflector topologywhen applied to the method of FIG. 4. Each of the areas that converge toa particular root can be represented by a color (i.e., in the drawings,differences in “color” can be generally shown by differing grayscaleshades). Each root has a different color associated with it. Thereforeareas represented by 260, 262, 264, 266, 268, and 270 converge to afirst root, while areas 272, 274, 276, 278, and 280 to a second root,282, 284, 286, 288, 290, and 292 converge to a third root, and finally,areas 294, 296, 298, and 300 converge to a fourth root. The interfacesbetween two different areas corresponding to two different roots isinfinitely small and cannot be construed an area in the x+jy space. Inother words, there are no areas in the x+jy space that do not convergeto one of the four roots of the function f(x). As a result, thisfunction is not a forcing function that will produce a convergentreflector topology.

Unlike FIG. 9, FIG. 10 shows an example of a Newton-Raphson fractal of afunction with areas of non-convergence. The areas defined by 360, 362,364, 366, 368 are areas where all points converge to a first root whenany point in those areas are applied to a convergence algorithm, such asa Newton-Raphson convergence algorithm. In other words, any pointapplied to a convergent algorithm in these areas will produce anotherpoint that converges to the same root and therefore that point is in anarea designated by the same color as the areas 360, 362, 364, 366, and368. The areas 370, 372, 374, 376 similarly contain points that convergeto a second root when applied to a convergent algorithm. Points in theareas 380, 382, 384, 386, 388, 390, and 392 do not converge to a root.Therefore there are areas that do not converge to a root for thisfunction. As a result, this function can be used as a forcing functionin the method for determining a convergent reflector topology of FIG. 4.

FIGS. 11A-D show examples of various laser emissions. FIG. 11A shows aradiation emission using reflector topologies known in the prior art.The radiation emission 432 from the partial reflectance reflector 422 isdistributed over the full surface of the reflector generating pattern434. A convergence of the radiation to specific regions is not observedin this prior art reflector design. On the other hand FIGS. 11B-D showexamples of convergent laser emissions. Spatial convergence of theemitted radiation does not have to be at the optical axis, but can beanywhere along the topography of the reflector. Additionally, based onthe forcing function that has been chosen, there may be multipleconvergence points. FIG. 11B shows a radiation of emission through thepartial reflectance reflector 422 with two convergent points 442 and444. As a result the emission pattern has an outside ring convergence446 with an internal point convergence 448. FIG. 11C shows anotherconvergent emission pattern with an outside ring 466 and inside area ofconvergence 468 resulting from emissions 462 and 464 through the partialreflectance reflector 422. FIG. 11D illustrates yet another convergentemission pattern where there is an annular region 486 of convergencewith a round area 488 of convergence inside of the annular region 486resulting from emissions defined by 482, 483, and 484.

All of the radiation patterns in FIGS. 11B-D have at least two points ofconvergence. In some cases, it may be advantageous to have a singlepoint of convergence, such as a single center point. In such a case, onemay design reflectors 420 and 422 to produce a radiation pattern asdepicted in any of FIGS. 11B-D and then modify portions of thereflectors 420 and 422, such that the rays 442, 462, 482, and 483 thatare not emitted from the origin point are reflected back in a manner sothat they emit as a ray from the origin point.

Referring now to FIGS. 12A and 12B, the concept of single point emissionis described in greater detail. Using the method of determining asurface topology of a reflector surface 200, as described above, totalreflectance reflector 520 and partial reflectance reflector 522 can bedesigned to produce a beam pattern with a center emission 544A and edgeemissions 542 through the partial reflectance reflector 522. Such anemission pattern is generated via multiple reflections within the cavitydepicted as 540A at multiple points P1-P9 on the total reflectancereflector 520 and partial reflectance reflector 522.

The partial reflectance reflector 522 as defined by surface topologydetermined from method 200 can further be partitioned into a firstsection 524 and a second section 526A. In the first section 522 does nothave the undesired edge emissions 542 emitted therethrough and thesecond section 526A does have the undesired edge emissions 542 emittedtherethrough. As a next step, the slope in the second section 526A ismodified in a manner such that any beam incident upon it will bereflected back from a 90° incident surface so that the beam is reflectedback to its origination point. By doing so, the second section 526A ismodified to second section 526B (FIG. 12B) where there is no emissionthrough the second section 526B. Furthermore the reflected rays aremodified to 540B, different from 540A, and all of the radiationreflected from the second section is ultimately emitted as part ofcenter emission 544B, which differs, at least in intensity from centeremission 544A.

The method 600 of generating a single convergent emission point is shownin FIG. 13. Initially, a convergent surface topology of the reflectorsurface must be determined using method 200. Next the reflector surfacemust be divided into two or more sections at 602. The sections must benon-overlapping and one section must contain the point from whereemission is desired and another section must contain the point or regionfrom where emission is not desired. Next, the slopes at the surface ofat least one of the portioned sections are modified at 604. The slopesat the surface of the modified section are set such that any lightincident upon that surface will be reflected back to the source of theincident light. The modified topology with the modified section is thentested or simulated at 606 to determine if it produces the desiredradiation pattern with a single convergence point. If it does not, theslopes in the modified section are further modified, so that incidentlight on any surface in that section is sufficiently reflected back tothe origin of the light. If however it is determined that that thedesired radiation pattern is achieved at 606, then the final topology ofthe reflectors are known at 608.

While the invention has been specifically described in connection withcertain specific embodiments thereof, it is to be understood that thisis by way of illustration and not of limitation. Reasonable variationand modification is possible within the scope of the forgoing disclosureand drawings without departing from the spirit of the invention which isdefined in the appended claims.

What is claimed is:
 1. A laser device, having an optical cavitycontaining a gain medium, a total reflectance reflector positionedwithin the optical cavity, and a partial reflectance reflectorpositioned within the optical cavity in a juxtaposed relationship to thetotal reflectance reflector, comprising: wherein the total reflectancereflector has a first surface topology; wherein the partial reflectancereflector has a second surface topology; wherein at least one of thefirst surface topology and the second surface topology is defined by aconvergent reflector topology function that converges light emittedwithin the optical cavity to a laser beam that exits the optical cavitythrough the partial reflectance reflector by a series of reflectionsbetween the total reflectance reflector as defined by the first surfacetopology and the partial reflectance reflector as defined by the secondsurface topology; and wherein the laser beam emitted from the opticalcavity has a predefined pattern for a given first surface topology and asecond surface topology.
 2. The laser device of claim 1 wherein thepredetermined pattern is a singular point.
 3. The laser device of claim1 wherein the predetermined pattern is at least one point.
 4. The laserdevice of claim 1 wherein the predetermined pattern is a ring thatcircumscribes a point.
 5. The laser device of claim 1 wherein the firstsurface topology is the same topology as the second surface topology, ina juxtaposed relationship to the second surface topology.
 6. The laserdevice of claim 1 wherein the laser beam exiting the partial reflectancereflector maintains the predefined pattern without requiring an opticalelement outside the laser cavity.
 7. The laser device of claim 1 whereinthe convergent reflector function of at least one of the first surfacetopology and the second surface topology is derived from a forcingfunction in a complex space having at least two unsolvable areas by rootapproximation methods.
 8. The laser device of claim 7 wherein theconvergent reflector function of at least one of the first surfacetopology and the second surface topology comprises a determined boundaryedge of each of the unsolvable areas.
 9. The laser device of claim 8wherein the convergent reflector function of at least one of the firstsurface topology and the second surface topology is derived from aniterated convergence algorithm beginning with a preselected startingpoint to create a stepping pattern in the complex space in the at leastone unsolvable area.
 10. The laser device of claim 9 wherein theconvergent reflector function of at least one of the first surfacetopology and the second surface topology is derived from a convertedstepping pattern in the real domain to form the at least one of thefirst topology and the second topology.
 11. The laser device of claim 1wherein the partial reflectance reflector has a higher reflectanceregion outside an area where the laser beam exits the cavity.
 12. Thelaser device of claim 11 wherein the partial reflectance reflectorcomprises a total reflectance region outside an area where the laserbeam exits the cavity.